1 Scope of the Chapter

This chapter is concerned with computing the zeros of a polynomial with real or complex coefficients.

2 Background to the Problems

Let fz be a polynomial of degree n with complex coefficients ai:

fz≡a0zn+a1zn-1+a2zn-2+⋯+an-1z+an, a0≠0.

A complex number z1 is called a zero of fz (or equivalently a root of the equationfz=0), if

fz1=0.

If z1 is a zero, then fz can be divided by a factor z-z1:

fz=z-z1f1z

(1)

where f1z is a polynomial of degree n-1. By the Fundamental Theorem of Algebra, a polynomial fz always has a zero, and so the process of dividing out factors z-zi can be continued until we have a complete factorization of fz:

fz≡a0z-z1z-z2…z-zn.

Here the complex numbers z1,z2,…,zn are the zeros of fz; they may not all be distinct, so it is sometimes more convenient to write

fz≡a0z-z1m1z-z2m2…z-zkmk, k≤n,

with distinct zeros z1,z2,…,zk and multiplicities mi≥1. If mi=1, zi is called a simple or isolated zero; if mi>1, zi is called a multiple or repeated zero; a multiple zero is also a zero of the derivative of fz.

If the coefficients of fz are all real, then the zeros of fz are either real or else occur as pairs of conjugate complex numbers x+iy and x-iy. A pair of complex conjugate zeros are the zeros of a quadratic factor of fz, z2+rz+s, with real coefficients r and s.

Mathematicians are accustomed to thinking of polynomials as pleasantly simple functions to work with. However, the problem of numerically computing the zeros of an arbitrary polynomial is far from simple. A great variety of algorithms have been proposed, of which a number have been widely used in practice; for a fairly comprehensive survey, see Householder (1970). All general algorithms are iterative. Most converge to one zero at a time; the corresponding factor can then be divided out as in equation (1) above – this process is called deflation or, loosely, dividing out the zero – and the algorithm can be applied again to the polynomial f1z. A pair of complex conjugate zeros can be divided out together – this corresponds to dividing fz by a quadratic factor.

Whatever the theoretical basis of the algorithm, a number of practical problems arise; for a thorough discussion of some of them see Peters and Wilkinson (1971) and Chapter 2 of Wilkinson (1963). The most elementary point is that, even if z1 is mathematically an exact zero of fz, because of the fundamental limitations of computer arithmetic the computed value of fz1 will not necessarily be exactly 0.0. In practice there is usually a small region of values of z about the exact zero at which the computed value of fz becomes swamped by rounding errors. Moreover, in many algorithms this inaccuracy in the computed value of fz results in a similar inaccuracy in the computed step from one iterate to the next. This limits the precision with which any zero can be computed. Deflation is another potential cause of trouble, since, in the notation of equation (1), the computed coefficients of f1z will not be completely accurate, especially if z1 is not an exact zero of fz; so the zeros of the computed f1z will deviate from the zeros of fz.

A zero is called ill-conditioned if it is sensitive to small changes in the coefficients of the polynomial. An ill-conditioned zero is likewise sensitive to the computational inaccuracies just mentioned. Conversely a zero is called well-conditioned if it is comparatively insensitive to such perturbations. Roughly speaking a zero which is well separated from other zeros is well-conditioned, while zeros which are close together are ill-conditioned, but in talking about ‘closeness’ the decisive factor is not the absolute distance between neighbouring zeros but their ratio: if the ratio is close to one the zeros are ill-conditioned. In particular, multiple zeros are ill-conditioned. A multiple zero is usually split into a cluster of zeros by perturbations in the polynomial or computational inaccuracies.